Relatively hyperbolic groups with free abelian second cohomology

Abstract

Suppose G is a finitely presented group that is hyperbolic relative to P a finite collection of finitely presented proper subgroups of G. Our main theorem states that if the second cohomology group H2(P,ZP) is free abelian for each P∈P, then H2(G,ZG) is free abelian. The problem reduces to the case when G and the members of P are 1-ended. In that case, we prove that a “Cusped Space” for this pair has semistable fundamental group at ∞. This provides a starting point in our proof of the main theorem. Finally we prove a proper simplicial approximation result for maps of [0,∞)×[0,1]. This result and our cusped space result are primary components in the proof of our main theorem. Both the cusped space result and the simplicial approximation result have applications beyond this article.